In recent work, we have shown that fundamental quantities such as the total entropy production, the thermodynamic variables conjugate to the thermodynamic forces, and the Glansdorff-Prigogine's dissipative variable may be discretized at the mesoscopic scale. The Canonical Commutation Rules (CCRs) valid at the mesoscopic scale have been postulated and the measurement process consists of determining the eigenvalues of the operators associated with the thermodynamic quantities. The essence of that work was to analyze the consequences of these postulates. This letter aims to understand the physical nature of the new constant (beta) introduced in the CCRs. In particular, two questions are still open. Is beta a new fundamental constant? Does the constant in the CCRs correspond to the lowest limit? We shall tackle the problem starting from the simplest assumption: the expression of beta can be derived from a simple model (a heuristic model) for nano-gas and studied through the tools of classical statistical physics. We will find that the theoretical value is very close to the experimental one. We shall not go further in our interpretation; we shall limit ourselves simply by noticing that according to our model, the constant beta does not appear to be a new fundamental constant but corresponds to the minimum value.
In recent work, we have shown that fundamental quantities such as the total entropy production, the thermodynamic variables conjugate to the thermodynamic forces, and the Glansdorff-Prigogine's dissipative variable may be discretized at the mesoscopic scale. The Canonical Commutation Rules (CCRs) valid at the mesoscopic scale have been postulated and the measurement process consists of determining the eigenvalues of the operators associated with the thermodynamic quantities. The essence of that work was to analyze the consequences of these postulates. This letter aims to understand the physical nature of the new constant (beta) introduced in the CCRs. In particular, two questions are still open. Is beta a new fundamental constant? Does the constant in the CCRs correspond to the lowest limit? We shall tackle the problem starting from the simplest assumption: the expression of beta can be derived from a simple model (a heuristic model) for nano-gas and studied through the tools of classical statistical physics. We will find that the theoretical value is very close to the experimental one. We shall not go further in our interpretation; we shall limit ourselves simply by noticing that according to our model, the constant beta does not appear to be a new fundamental constant but corresponds to the minimum value.
This letter aims to investigate thermodynamic processes in small systems in the Onsager region by showing that fundamental quantities such as total entropy production can be discretized on the mesoscopic scale. Even thermodynamic variables can conjugate to thermodynamic forces, and thus, Glansdorff–Prigogine’s dissipative variable may be discretized. The canonical commutation rules (CCRs) valid at the mesoscopic scale are postulated, and the measurement process consists of determining the eigenvalues of the operators associated with the thermodynamic quantities. The nature of the quantized quantity β , entering the CCRs, is investigated by a heuristic model for nano-gas and analyzed through the tools of classical statistical physics. We conclude that according to our model, the constant β does not appear to be a new fundamental constant but corresponds to the minimum value.
In previous work, we investigated thermodynamic processes in systems at the mesoscopic level where traditional thermodynamic descriptions (macroscopic or microscopic) may not be fully adequate. The key result is that entropy in such systems does not change continuously, as in macroscopic systems, but rather in discrete steps characterized by the quantization constant β. This quantization reflects the underlying discrete nature of the collision process in low-dimensional systems and the essential role played by thermodynamic fluctuations at this scale. Thermodynamic variables conjugate to the forces, along with Glansdorff-Prigogine's dissipative variable can be discretized, enabling a mesoscopic-scale formulation of canonical commutation rules (CCRs). In this framework, measurements correspond to determining the eigenvalues of operators associated with key thermodynamic quantities. This work investigates the quantization parameter β in the CCRs using a nano-gas model analyzed through classical statistical physics. Our findings suggest that β is not an unknown fundamental constant. Instead, it emerges as the minimum achievable value derived from optimizing the uncertainty relation within the framework of our model. The expression for β is determined in terms of the ratio χ, which provides a dimensionless number that reflects the relative scales of volume and mass between entities at the Bohr (atomic level) and the molecular scales. This latter parameter quantifies the relative influence of quantum effects versus classical dynamics in a given scattering process.
In this work, improvements are introduced to the current models of the ideal Fermi gas and the ideal Bose gas by incorporating the quantum nature of phase space, which is directly linked to the uncertainty principle. These improved models build upon the recently developed concepts of quantum phase space (QPS) and the QPS representation of quantum mechanics. The Hamiltonian operator for a gas particle and its eigenstates are first determined, and quantum statistical mechanics is used to derive the thermodynamic properties of the ideal gas. Analytic expressions for thermodynamic quantities—including the grand canonical potential, particle number, internal energy, von Neumann entropy, and pressure—are derived, along with the corresponding thermodynamic equations of state for both bosons and fermions. These corrections are particularly significant at low temperatures and in confined volumes, where quantum effects related to system geometry, such as shape and size, become significant. The results also establish a direct link between thermodynamic functions and the quantum statistical variances of momentum. Importantly, the improved models recover well-known classical relations of the ideal gas in the high-temperature and large-volume limits, ensuring consistency with classical physics. By addressing quantum corrections and their thermodynamic implications, this work provides a foundation for further applications in nanoscale systems, quantum gases, low-temperature physics, ultracold physics, and astrophysics.
Abstract Quantum thermodynamics describes dynamic processes by means of the operators of entropy production P and time t . P and t do not commute. It exists the non-vanishing t–P commutator [ t , P ]=ik. Here the Boltzmann constant k has the physical meaning of a quantum of entropy. The t–P commutator immediately leads us to the t–P uncertainty relation ΔtΔP≥k/2. Hence the observables t and P are not sharply defined simultaneously. Similar uncertainty relations can also be expected for other pairs of conjugate variables with products of the physical meaning of an entropy. The free energy F and the reciprocal temperature (1/T) are the respective conjugate variables of an isolated system of many particles, which leads us to the F–(1/T) uncertainty relation ǀΔFǁΔ(1/T)ǀ≥k/2. It can be traced back to the t–P uncertainty relation mentioned above. In this way the Helmholtz free energy F and the temperatur T are introduced into quantum thermodynamics. The uncertainties ǀΔFǀ→0 and ǀΔTǀ→0 are negligible at low temperatures T→0, and quantum thermodynamics turns into the time-independent classical thermodynamics. Against this the uncertainties ǀΔFǀ→∞ and ǀΔTǀ→∞ grow unlimited at high temperatures T→∞, and classical thermodynamics loses its sense. In the limit of one particle the uncertainties cannot be neglected even at low temperatures. However a detailed discussion shows that the free energy f of a single particle vanishes within the whole range of temperatures T. This defines the particle entropy σ=ε/T=ak. The dimensionless entropy number a connects the particle energy ε=akT with the temperature T. The entropy number a of a single (s) independent particle can be calculated with the extended, temperature-dependent Schrödinger equation A s φ=aφ. Here A s=−(Λ 2/4π)∇2 means the dimensionless entropy operator describing the entropy number a and thus the particle entropy σ=ak. Λ is the thermal de Broglie wave length. Finally we calculate by means of quantized particle entropies σ the internal energy E, the Helmholtz free energy F, the entropy S, the chemical potential μ, and the equation of state of an ideal gas of N monatomic free particles in full agreement with classical thermodynamics. We also calculate the partition function q=V/Λ 3 of a single free particle within the volume V. Here Λ 3 is a small volume element taking into account the wave–particle dualism of a single free particle of mass m at temperature T. Extension to a system of N free particles leads us to a simple geometrical model and to the conclusion that an ideal gas of independent particles becomes instable below a critical temperature T C. T C corresponds to the critical temperature T BE of Bose–Einstein condensation.
The proper definition of entropy is fundamental to the relationship between statistical mechanics and thermodynamics. It also plays a major role in the recent debate about the validity of the concept of negative temperature. In this paper, I analyze and calculate the thermodynamic entropy for large but finite quantum mechanical systems. A special feature of this analysis is that the thermodynamic energy of a quantum system is shown to be a continuous variable, rather than being associated with discrete energy eigenvalues. Calculations of the entropy as a function of energy can be carried out with a Legendre transform of thermodynamic potentials obtained from a canonical ensemble. The resultant expressions for the entropy are able to describe equilibrium between quantum systems having incommensurate energy-level spacings. This definition of entropy preserves all required thermodynamic properties, including satisfaction of all postulates and laws of thermodynamics. It demonstrates the consistency of the concept of negative temperature with the principles of thermodynamics.
Thermodynamic uncertainty relations (TURs) represent one of the few broad-based and fundamental relations in our toolbox for tackling the thermodynamics of nonequilibrium systems. One form of TUR quantifies the minimal energetic cost of achieving a certain precision in determining a nonequilibrium current. In this initial stage of our research program, our goal is to provide the quantum theoretical basis of TURs using microphysics models of linear open quantum systems where it is possible to obtain exact solutions. In paper [Dong et al., Entropy 2022, 24, 870], we show how TURs are rooted in the quantum uncertainty principles and the fluctuation-dissipation inequalities (FDI) under fully nonequilibrium conditions. In this paper, we shift our attention from the quantum basis to the thermal manifests. Using a microscopic model for the bath's spectral density in quantum Brownian motion studies, we formulate a "thermal" FDI in the quantum nonequilibrium dynamics which is valid at high temperatures. This brings the quantum TURs we derive here to the classical domain and can thus be compared with some popular forms of TURs. In the thermal-energy-dominated regimes, our FDIs provide better estimates on the uncertainty of thermodynamic quantities. Our treatment includes full back-action from the environment onto the system. As a concrete example of the generalized current, we examine the energy flux or power entering the Brownian particle and find an exact expression of the corresponding current-current correlations. In so doing, we show that the statistical properties of the bath and the causality of the system+bath interaction both enter into the TURs obeyed by the thermodynamic quantities.